A Trotter-type approach to infinite rate mutually catalytic branching

TL;DR

This paper introduces a Trotter-type construction for the infinite rate mutually catalytic branching process, providing a new method to analyze its properties and long-term behavior, complementing previous noise equation approaches.

Contribution

It presents a novel Trotter-type construction for IMUB, enabling new analysis techniques for its long-time behavior and coexistence or segregation of types.

Findings

Constructed IMUB via Trotter approach

Facilitates analysis of long-term behavior

Supports future studies on coexistence versus segregation

Abstract

Dawson and Perkins [Ann. Probab. 26 (1988) 1088--1138] constructed a stochastic model of an interacting two-type population indexed by a countable site space which locally undergoes a mutually catalytic branching mechanism. In Klenke and Mytnik [Preprint (2008), arXiv:0901.0623], it is shown that as the branching rate approaches infinity, the process converges to a process that is called the infinite rate mutually catalytic branching process (IMUB). It is most conveniently characterized as the solution of a certain martingale problem. While in the latter reference, a noise equation approach is used in order to construct a solution to this martingale problem, the aim of this paper is to provide a Trotter-type construction. The construction presented here will be used in a forthcoming paper, Klenke and Mytnik [Preprint (2009)], to investigate the long-time behavior of IMUB (coexistence versus segregation of types). This paper is partly based on the Ph.D. thesis of the second author (2008), where the Trotter approach was first introduced.